I run my gas engines dry for seasonal storage, but should I do the same with my new summer-use diesel (3-cyl Kubota), or just leave it with a full tank? If it should get a fuel additive first, what kind (not what brand, because not all are sold everywhere)?

I only use our tractor in the winter for snow. Some years we don't get any snow and I've never had a problem just leaving the tank full with a fuel preservative and anti-gel diesel fuel treatment with biocide (I think that's what it's called). Every month or so it is started and run at a fast idle for a half hour to keep it loose and lubed and the battery charged.

Absolutely leave it with a full tank (of treated fuel). I fill mine up nearly every time I finish using it for the day, no matter what time of year it is. A full tank doesn't expose any inside tank walls for condensation to collect.

As far as additives, click the Froogle link below. Personally, I use Power Service; white jug during the winter, silver jug the rest of the year.